where \({Z}_{1}\)  and \({Z}_{2}\)  are vector Gaussian random variables with positive definite covariance matrix, and \(\mu \)  is any real number greater than 1. In [1]}, it is shown that the extremal auxiliaries of (REF ) are Gaussian distributed. In the proof of extremal inequality, two key techniques are involved. One is the channel enhancement argument [2]}, which is exploited to convert \({Z}_{1}\)  and \({Z}_{2}\)  in degraded order. The other is a strengthened perturbation approach following by Stam [3]} and Blachman [4]} in the proof of entropy power inequality (EPI) [5]}.
